Current

Modeling and simulation of tissues using high resolution cell models in 3D

Mathematical models are increasingly designed to guide experiments in biology, biotechnology, as well as to assist in medical decision making. For exmaple, we may investigate the mechanical stresses experienced by the hepatocytes during liver tissue regeneration , and clarify the relation with resulting tissue architecture. Over the past couple of years, I have been elaborating on a high resolution "Deformable Cell Model" (DCM) in 3D (see Fig. 1) that is capable of representing cell shape and integrating information at subcellular scales [1,3] (PDF). In a basic DCM the cell surface is discretized by a number of nodes which are connected by viscoelastic elements, creating a flexible scaffolding structure with arbitrary degrees of freedom per cell. The model can used to simulate in vitro single cell experiments in order to extract information at subcellular scales. The triangulated structure further allows cells to interact with arbitrarily shaped structural elements. We have extended the model with the capability for the cells to grow and divide so that realistic tissue simulations are possible. I am currently using this model to investigate early embryonic development. This work is in collaboration with prof. Frédéric Lemaigre, UCL, Belgium.

Fig.1b: High resolution cell model used in a simulation of hepatocyte regeneration in a liver lobule. The cells grow through the network of blood vessels (depicted in gray) from the portal vein area towards the central vein

The relation between cell growth and mechanical stress

This work aims to clarify the bio-mechanical effects of growing tumor cells in confining surrounding stroma. Using agent-based models, I studied growth dynamics of tumor cells (proliferation, growth, necrosis) under different mechanical pressure conditions. One multi-cellular spheroid is growing in an elastic capsule, the other one in a Dextran solution (see Fig. 2). To simulate hypotheses concerning cell growth under pressure conditions, the model quantitatively takes into account the applied mechanical stress, cell density, and growth rate. In this work, it was demonstrated that the cellular growth response on external mechanical stress may be surprisingly quantitatively predictable over various cell lines, independently of the environmental conditions [2].

*Fig.2: We address two different experiments to investigate growing spheroids under pressure.*

Modeling of cell-ECM interactions

The influence of mechanical feedback loop between cells and Extra-Cellular Matrix (ECM) , the medium in which cells anchor, is poorly understood. In collaboration with Dr. Andreas Buttenschoen (University of British Colombia), Dr. Margriet Palm (U. of Leiden) and Tommy Heck (KULeuven) we are creating models for cells embedded in ECM material. The cells have anchoring points (filopodia) to the ECM. The ECM is either modeled by a continuum material or a discrete network of viscoelastic elements. By a sequence of pulling and retracting the anchors, the cells move through the ECM. As such we can quantify both the stresses that work on the cells and the ECM. Part of this work is published in [4].

Past (before 2013)

Development of high resolution ”Deformable Cell Model“: modeling of arbitrary shaped cells with high detail

The majority of agent based models in the past were build on the assumption of a rigid shaped cells (so-called center based models: e.g. spheres, ellipsoids). In a some problems, such simplifications suffice to accurately simulate the biological or clinical problem (e.g. tumor growth). Yet, as more and more detail about cell shape and bio-mechanical forces is getting required nowadays, the development of more complex models has become timely.

*Fig.3: Simulation of a cell adhering to a surface (image from [3]).*

In collaboration with my former collegues in KULeuven, we introduced a so-called Deformable Cell Model (DCM) . This model type therefore allows simulations where cells shape and subcellular detail are important variables. In a first step, we studied the adhesion dynamics of cells to a surface. The developed contact adhesion model is a discretization of the Maugis-Dugdale adhesion theory. The model gives very good results with respect to experiments and lead to several new insights [3]. The principals of this contact model can also be further exploited to study contact mechanics of irregularly shaped rigid bodies in DEM [10].

Modeling of red blood cells in Stokes flow

Smoothed Particle Hydrodynamics (SPH) is a Lagrangian meshfree particle based method to simulate Navier-Stokes dynamics involving free surfaces, large deformations and complex physics. However, processes at cellular scale are usually viscosity dominated (overdamped) and hence a Stokes regime can be assumed. We developed an SPH solver for fluids in Stokes regime. The method (called NSPH) is based on the standard SPH technique but reduces it to a first order system and involves a different numerical solving scheme [5]. Thanks to this, longer timescales can be simulated, typically needed in cellular processes. I applied this method to the dynamics of a red blood cell passing a through a narrow channel (capillary). The blood plasma was modeled by NSPH, while the red blood cell was represented by a connection of elastic elements (see Fig.4).

*Fig.4: Simulation of Stokes dynamics using SPH: red blood cell passing through a narrow capillary.*

Modeling of viscoelastic bio-fluids

This project was a bilateral collaboration between KULeuven and the company MOBA N.V (The Netherlands) to understand and optimize the yield of albumen in industrial egg breaking machines. This is a very complicated process as the egg contents contain non- Newtonian fluids, membranes and complex geometries. I implemented a Smoothed Particle Hydrodynamics model for simulating viscoelastic fluids (Maxwell/Giesekus constitutive models) including surface tension. The final model encompassed a 3D realistic egg shell geometry with inside a distinct egg yolk and albumen.

Modeling of impact of cellular tissues and multi-scale modeling of cellular mechanics in visco-elastic tissues.

During my late PhD and early post-doc years, I gained interest in simulation of living matter, i.e cells and tissue. I became involved in a multiscale project and collaboration between two KUL reseach groups. The goal in this project was to understand how cellular tissue responds to (dynamic) mechanical load. Hereby, the project aimed at providing insights on how two numerical methodologies in mechanical modeling of tissue, whereby one is interested in linking microscopic properties (cellular level) to macroscopic (tissue-level) behavior, can be cast into an integrative approach. This work was a scientifically very successful collaboration between the Mebios lab (Faculty of Bio-engineering, KULeuven) and the department of applied mathematics (Faculty of Engineering, KULeuven). The tissue-level is assumed to behave viscoelastic and simulated using nonlinear Finite Element Methods, while the cellular scale is approached by particle based methods and incorporated the subcellular features such as cytoplasm viscosity, cell shape, etc.

*Fig.5: Snapshot of simulated 2D cellular structures and 3D cell during impact. Each cell is represented by a number of particles describing liquid and solid phases.*

The coupling between the two scales was achieved using computational homogenization techniques and Representative Volume Elements (RVEs) [7,8]. The microscale model involves a physical representation of cells and their parts (membranes, cytoplasm, nucleus, ..). The cells interact with each other by adhesive forces. Small groups of cells form a aggregate that represents a small but repetitive part of the tissue. The model is able to simulate large deformations and even breakage of cells under impact load (see Fig. 5) [6]. This project resulted in a working code that uses microscopic physical properties of cells (e.g. cytoplasm viscosity, cell wall Young modulus, cellular structure) and predicts the overall constitutive behavior of the resulting tissue within a feasible computation time [6]. Several innovative contributions to the field in order to accomplish the mission of this project had to be introduced: + Construction and implementation of a detail mechanical model of a biological cell with flexibility towards physical features and high accuracy. + Introduction of viscous inter-particle forces posed challenging problems with regard to the standard computational homogenization strategies. However, we successfully developed a technique to incorporate these effects in the boundary value problem for the Representative Volume Element.

PhD work: modeling of granular material flow using the Discrete Element Method

Discrete Element Method (DEM) simulations have become standard tools to simulate granular matter flow dynamics, because of increasing interest (pharmacy, mining industry, ..) and improved com- putational means. Performing DEM simulations requires a extensive knowledge on contact mechanics, multi-body dynamics, as well as efficient contact detection algorithms, and C++ coding. Model (DEM). During my PhD, I studied dynamics and contact mechanics of rigid materials in bulk, in particular the collective motion of grains and the interaction with rigid machine parts. This involved the development and application of a particle-based code. The platform Mpacts (http://dem-research-group.com) is now a mature C++/python oriented modular code to simulate particle dynamics. It is capable of simulating solid dynamics (grains, powders, arbitry shaped objects,...) using the Discrete Element Method, as well as com- plex fluid problems (Newtonian, non-Newtonian) using Smoothed Particle Hydrodynamics. In the early years of 2000, our the KULeuven research group was one the first groups to apply this method to industrial problems. Nowadays, the software is used by several companies.

*Fig.6: PhD work, development of a DEM framework to predict and optimize granular flow in agricultural processes.*

The goal of my PhD thesis was to build a numeri- cal model to simulate and quantify granular flow and the spread of grain particles in a spinning disc fertilizer spreader, and thus to increase the optimiza- tion of the performance of agricultural machines (see Fig. 6). During the first years (2001-2005), I mainly developed the model and code for interacting grains including spherical and non-spherical shapes, and the interaction with rotating machine parts. In order to validate the models and calibrate model parameters, I built several experiments setups (e.g. [9]). From the insight obtained with simulations, we developed a working code that predicts fertilizer particle spreading on a field with a reasonable accuracy. The project was a successful collaboration with the Institut national de recherche en sciences et technologies pour l’environnement et l’agriculture (IRSTEA, Montoldre, France), the Institute for Agricultural and Fisheries Research in Belgium (ILVO), and several industrial partners. In the period 2006-2007, I also advised the company BASF in developing a prototype of a new fertilizer spreader machine (BASF, US patent 6588685). This was within the framework of a one year bilateral project.

References

1. Van Liedekerke P., J. Neitsch, T. Johann, E. Warnt, S. Hoehme, S. Grosser, J. Kaes and D. Drasdo (2017) Quantifying the mechanics and growth of cells and tissues in 3D using high resolution computational models (submitted). PDF
2.Van Liedekerke P., J. Neitsch, T. Johann, K. Alessandri, P. Nassoy and D. Drasdo (2017) Quantitative modeling identifies robust predictable stress response of growing CT-26 tumor spheroids under variable conditions (Plos comp. Biol, accepted).
3. T. Odenthal, B. Smeets, Van Liedekerke P., E. Tijskens, H. Ramon, H. Van Oosterwijck (2013) Contact mechanics of adhesive triangulated bodies and application to a deformable cell model. Plos Comp. Biol9(10).
4. T. Heck, S. Vanmaercke, P. Bhattacharya, T. Odenthal, H. Ramon, H Van Oosterwijck and Van Liedekerke P. (2017) Modeling ExtraCellular matrix viscoelasticity using non-inertial Smoothed Particle Hydrodynamics. Comp. Meth. Appl. Mech. Eng. 322(1).
5. Van Liedekerke P., T. Odenthal, B. Smeets, H. Ramon (2013) Solving microscopic flow problems using Stokes equations in SPH. Computer Physics Communications 184(7).
6. Van Liedekerke P., Ghysels P., Tijskens E., Samaey G., Roose D. and Ramon H. (2011) The bruising of soft cellular tissue: a particle base simulation approach. Soft Matter 7, DOI:10.1039/C0SM01261K.
7. Ghysels P., Samaey G., Tijskens B., Van Liedekerke P., Ramon H. and Roose D. (2009) Multi-scale simulation of plant tissue deformation using a model for individual cell mechanics. Phys. Biol. 6(3).
8. Ghysels P., Samaey G., Van Liedekerke P., Tijskens E., Ramon H. and Roose D. (2010) Multi-scale modeling of viscoelastic plant tissue. Int. J. Multiscale Com. Eng. 8(4).
9. Van Liedekerke P., Tijskens E., Dintwa E.,F. Rioual, J. Vangeyte and Ramon H. (2008) DEM simulations of the particle flow on a centrifugal fertilizer spreader. Powder Technology 190(3), 348-360.
10. B. Smeets,T. Odenthal, J. Keresztes, S. Vanmaercke, Van Liedekerke P., E. Tijskens, W. Saeys, H. Ramon, H. Van Oosterwijck (2014) Modeling contact interactions between triangulated rounded bodies for the discrete element method. Comp. Meth. Appl. Mech. Eng. 275(10).

Paul Van Liedekerke,

PhD, HDR